Quotes from John von Neumann

When we talk mathematics, we may be discussing a secondary language built on the primary language truly used by the central nervous system.

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.

Natura non facis Saltus.

In the deceptively modest volume you are now holding, von Neumann articulates his model of computation and goes on to define the essential equivalence of the human brain and a computer. He acknowledges the apparently deep structural differences, but by applying Turing's principle of the equivalence of all computation, von Neumann envisions a strategy to understand the brain's methods as computation, to re-create those methods, and ultimately to expand its powers.

Since I am neither a neurologist nor a psychiatrist, but a mathematician, the work that follows requires some explanation and justification.

In an analog machine each number is represented by a suitable physical quantity, whose values, measured in some pre-assigned unit, is equal to the number in question.

Any computing machine that is to solve a complex mathematical problem must be 'programmed' for this task. This means that the complex operation of solving that problem must be replaced by a combination of the basic operations of the machine.

Thus all sorts of sophisticated order-systems become possible, which keep successively modifying themselves and hence also the computational processes that are likewise under their control.

The very last stage of any memory hierarchy is necessarily the outside world—that is, the outside world as far as the machine is concerned, i.e. that part of it with which the machine can directly communicate, in other words, the input and the output organs of the machine. These are usually punched paper tapes or cards, and on the output side, of course, also printed paper.

All existing machines and memories use "direct addressing," which is to say that every word in the memory has a numerical address of its own that characterizes it and its position within the memory (the total aggregate of all hierarchic levels) uniquely.

The linear size of a neuron varies widely from one nerve cell to the other, since some of these cells are contained in closely integrated large aggregates and have, therefore, very short axons, while others conduct pulses between rather remote parts of the body and may, therefore, have linear extensions comparable to those of the entire human body.

The human brain is, after all, the best example we have of an intelligent system. If we can learn its methods, we can use these biologically inspired paradigms to build more intelligent machines. This book is the earliest serious examination of the human brain from the perspective of a mathematician and computer pioneer. Prior to von Neumann, the fields of computer science and neuroscience were two islands with no bridge between them.

He notes that the output of neurons is digital: an axon either fires or it doesn't. This was far from obvious at the time, in that the output could have been an analog signal. The processing in the dendrites leading into a neuron and in the soma neuron cell body, however, are analog. He describes these calculations as a weighted sum of inputs with a threshold.

As he describes each mechanism in the brain, he shows how a modern computer could accomplish the same operation, despite the apparent differences. The brain's analog mechanisms can be simulated through digital ones because digital computation can emulate analog values to any desired degree of precision (and the precision of analog information in the brain is quite low).

Truth is much too complicated to allow anything but approximations.

It would appear that we have reached the limits of what it is possible to achieve with computer technology, although one should be careful with such statements, as they tend to sound pretty silly in 5 years.

Technological possibilities are irresistible to man. If man can go to the moon, he will. If he can control the climate, he will.

Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin.

Computers are like humans - they do everything except think.

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.